The generating function of the 𝑀₂-rank of partitions without repeated odd parts as a mock modular form

Jennings-Shaffer, Chris

doi:10.1090/tran/7212

The generating function of the $M_2$-rank of partitions without repeated odd parts as a mock modular form
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Trans. Amer. Math. Soc. 371 (2019), 249-277 Request permission

Abstract:

By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of this function as a harmonic Maass form and show more can be done with this function. In particular, we show the related harmonic Maass form transforms like the generating function for partitions without repeated odd parts (which is a modular form). We then use these improvements to determine formulas for the rank differences modulo $7$. Additionally, we give identities and formulas that allow one to determine formulas for the rank differences modulo $c$, for any $c>2$.

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